The Barry projectk-anonymity on a census table: how small is too small?
Count the households hiding in tiny cells, then blur the table two different ways and count again.
The learner downloads Census 2021 table TS017, household size, from the Nomis API for all 434 output areas in the Vale of Glamorgan. The area rows add up to 57,452 households sorted into eight sizes, from one person to eight or more. The published county total is 57,446, six fewer, and that small gap is itself a privacy measure, as explained below. Each combination of an area and a size is a cell. Two columns, where you live and how many live with you, are things a neighbour would know; privacy researchers call such columns quasi-identifiers. A household is exposed at level k if its cell holds fewer than k households.
Households in cells of fewer than five, under different levels of detail, our Python count on Census 2021 TS017 for the Vale of Glamorgan| Areas | Household-size groups | Cells under 5 | Households in them |
|---|
| 434 output areas | 8 sizes | 571 | 1,134 (1.97%) |
| 434 output areas | 1, 2, 3, 4, 5 or more | 146 | 430 (0.75%) |
| 434 output areas | 1, 2, 3 or more | 0 | 0 |
| 82 larger areas (LSOAs) | 8 sizes | 109 | 209 (0.36%) |
| 82 larger areas (LSOAs) | 1, 2, 3, 4, 5 or more | 0 | 0 |
At full detail, 571 of the 3,472 cells hold between one and four households, 1,134 households in all, and another 930 cells are empty. Of those 571 small cells, 555 are for households of five or more people: homes that size are scarce in any one area, so they are easy to pick out. There are two ways to blur. Merging the sizes into "five or more" cuts the exposed households to 430 but does not finish the job. Keeping all eight sizes and moving to larger areas leaves 209. Doing both, five groups across 82 areas, leaves no cell under five, and the smallest holds 12. So does keeping the small areas and using only three groups, but that throws away nearly everything the table said about larger households.
Raising k raises the price. At k of 10 the full-detail table has 854 small cells covering 3,033 households, 5.28% of the county. One caution belongs on the page: the ONS already adjusts small census counts slightly to protect people, which is why the rows add to 57,452 while the county total reads 57,446. These published numbers are the table a reader sees, not exact truths. The exercise is about how to reason over any table, including ones nobody has protected.
Ages 8 to 11
Play a guessing game: how few clues does it take to pick one classmate out of thirty?
Ages 11 to 15
Load the Vale household table in Python and list every cell that holds fewer than five.
Ages 15 and up
Measure k-anonymity at several levels of detail and chart what each level of safety costs.
Census table, our counting
Household counts are Office for National Statistics Census 2021 data from Nomis, under the Open Government Licence, and already carry the ONS's own disclosure protection. The cells, groupings and percentages are our calculations. Nothing here identifies any household, and no attempt was made to.
